To solve this problem, we need to analyze the forces acting on the bead when the wire is accelerated. The wire is shaped like a parabola, and the bead can slide along it without friction. When the wire is accelerated parallel to the x-axis, the bead will experience a new equilibrium position due to the effects of acceleration. Let's break down the steps to find the distance of the new equilibrium position from the y-axis.

Understanding the Forces at Play

Initially, when the wire is at rest, the bead is at the lowest point of the parabola, which we can denote as the origin (0,0) in a coordinate system where the vertex of the parabola is at the origin. The equation of the parabola can be expressed as:

y = kx²

where k is a constant that determines the shape of the parabola.

Effect of Acceleration

When the wire is accelerated to the right with a constant acceleration a, the bead will experience a pseudo-force acting to the left due to the non-inertial reference frame of the wire. This pseudo-force can be expressed as:

F_p = m(-a)

where m is the mass of the bead.

Finding the New Equilibrium Position

To find the new equilibrium position, we need to consider the balance of forces acting on the bead. The bead will move to a new position where the net force acting on it is zero. The forces acting on the bead are:

• The gravitational force acting downwards: F_g = mg
• The normal force from the wire acting perpendicular to the surface of the parabola.
• The pseudo-force due to the acceleration of the wire.

At the new equilibrium position, the bead will be at a point (x, y) on the parabola. The slope of the parabola at this point can be found by differentiating the equation:

dy/dx = 2kx

At equilibrium, the tangent to the parabola at the new position must balance the gravitational force and the pseudo-force. The angle θ that the tangent makes with the horizontal can be found using:

tan(θ) = dy/dx = 2kx

Using the balance of forces in the x-direction, we have:

mg sin(θ) = ma

Substituting for sin(θ) using the relation:

sin(θ) = tan(θ) / √(1 + tan²(θ))

we can express the equilibrium condition as:

mg (2kx) / √(1 + (2kx)²) = ma

Solving for x

Rearranging this equation gives:

g(2kx) = a√(1 + (2kx)²)

Squaring both sides and simplifying will lead us to a quadratic equation in terms of x. Solving this quadratic will yield the distance of the new equilibrium position from the y-axis.

Final Result

After performing the necessary algebra, you will find that the new equilibrium position x can be expressed in terms of the parameters of the system, specifically g, a, and k. The exact distance will depend on these values, but the approach outlined here provides a systematic way to derive the new position of the bead on the parabolic wire.

In summary, the new equilibrium position of the bead is determined by balancing the gravitational force and the pseudo-force due to the acceleration of the wire, leading to a new position along the parabola that can be calculated using the derived equations.